'Polycircles' printed from http://nrich.maths.org/

Why do this problem?
This problem is accessible for different age groups. It is a
rich problem that provides
suitable challenges for a wide age range. Younger learners can find
solutions for three circles given the sides of a triangles by trial
and improvement.

Once the learners can solve two simultaneous linear equations in
two unknowns then they can be challenged to solve this problem by
solving three simultaneous equations. Because the coefficients are
all unity the equations are easy to solve.

This problem is often used at stages 3 as well as 4. At Stage 5 it
is useful as an application of solving simultaneous linear
equations and the question about which polygons have solutions and
which do not can take learners into linear algebra.

The problem demonstrates a powerful inter-relationship between
geometry and algebra. It also links to the problem
Pentagon from 1926 which has a different context but where the
mathematics is identical giving an experience of isomorphism in
mathematics.

Possible approaches
For younger learners who know how to solve a pair of simultaneous
linear equations this problem provides a good series of challenges.
Taking a numerical example where the lengths of the sides of the
triangle are known (see the Hint), and the radii of the polycircles
has to be found, then three simultaneous linear equations can
easily be found and solved. Even though learners may only have been
taught to solve two simultaneous equations in two unknowns many
will be able to solve these three equations for themselves and get
satisfaction from being able to do so independently. The results
can be checked by drawing.

The next step is to generalise from a particular numerical example
and to use exactly the same steps in the algebra to derive formulae
for the radii in terms of the lengths of the sides of the
triangle.This is a good exercise in a concrete setting for working
with algebra.

Learners who have succeeded so far can explore the existence of
polycircles for quadrilaterals and pentagons and how the algebra
explains the different geometrical phenomena that arise.

For older learners who have been been introduced to linear algebra,
the generalisation of the problem to polygons with n sides provides
a challenge to explain why there are unique solutions for certain
values of n and not for others.

Possible support
The problem is written to start with a numerical special case of a
triangle in order to support younger and less confident
learners.They may first try trial and improvement where the problem
is like arithmagons. From that they may be able to develop a method
for finding solutions in the general case and even for getting the
formula without the use of simultaneous equations.

Possible extension
You could pose the problem for general polygons and leave the
learners to decide for themselves whether or not to start with
special cases.